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The Concept of -Cycle and Applications
The concept of -cycle is investigated for its properties and applications. Connections with irreducible polynomials over a finite field are established with emphases on the notions of order and degree. The results are applied to deduce new results about primitive and self-reciprocal polynomials.
Let denote the finite field of elements and let , . Let be distinct numbers chosen from . If then we say that forms a -cycle mod with leading element , abbreviated by -cycle or -cycle when the leading element in the cycle is immaterial, and call the length of this -cycle. The notion of -cycles was introduced by Wan in his book [1, page 203]. Since , where is the order of in (the multiplicative group of nonzero integers modulo ), it clearly follows that each -cycle always has a unique length which is the least positive integer for which . Observe that -cycles are nothing but -orbits ; that is, acts on by multiplication with . The concept of -cycles is important because of the following connections with irreducible polynomials.
(A) (see [1, Theorem 9.11]) Let be a primitive th root of unity (if the order of in is , then there exists a primitive th root of unity in ). If is a -cycle, then is a monic irreducible factor of in . Conversely, if is a monic irreducible factor of in , then all the roots of are powers of whose exponents form a -cycle. We henceforth refer to these two facts as the -correspondence.
(B) (see [1, Corollary 9.12]) The number of distinct irreducible polynomials dividing in is equal to the number of -cycles formed with the leading elements taken from .
Our objectives here are to illustrate the versatility of -cycles by using them to prove new results about irreducible polynomials. General properties of -cycles are given in the next section; specific details in two special cases corresponding to are worked out for applications to primitive and self-reciprocal polynomials in the last section. Section 3 deals with results about the order of a polynomial, while Section 4 does the same for the degree of a polynomial. Section 5 shows that knowing a -cycle is equivalent to knowing all coefficients of the corresponding polynomial (2). The last section provides applications of -cycles to primitive and self-reciprocal polynomials.
Notation and Terminology. Throughout, we fix the following symbols and their meanings.(i)is a fixed prime, is a power of , and is the finite field with elements.(ii) is a fixed positive integer such that .(iii) is the number of (positive integer) divisors of .(iv) (all divisors of ).(v)For , the Möbius function is defined by (vi)For , the Euler's function is the number of integers with .(vii)For , denote by the order of in ; that is, (viii)For , set , the set of all -cycles .
2. Properties of -Cycles
Let be a -cycle. It is easy to see that can be decomposed into a finite union of disjoint -cycles, namely, Since and for all , each -cycle is of length with the one containing having the largest length. We collect in our first theorem further properties of -cycles, whose straightforward proofs are omitted.
Theorem 1. Let , , . (i)The element -cycle if and only if for some .(ii)Two -cycles, -cycle and -cycle, are identical if and only if -cycle.(iii)For , each -cycle has length if and only if divides but does not divide (if , only the first divisibility needs to be checked). The -cycle has length 1.(iv)The length of each -cycle divides .(v)If is such that , then the -cycle has length , and there are altogether distinct such -cycles. In particular, if is such that , then the -cycle has length , and there are altogether distinct -cycles with length .(vi)The length of any -cycle is of the form for some .(vii)The number of -cycles of length is .(viii)The total number of -cycles is ( signifies the number of elements in the set).
2.1. Two Special Cases
There are two particular cases and , which are closely related to primitive and self-reciprocal and will be needed in the last section.
2.1.1. The Case
When , more precise information is now derived.
Theorem 2. Let . Then (i);(ii) is the largest length among -cycles and any other length of -cycles divides ;(iii)the number of -cycles having length is equal to
Proof. Assertions (i) and (ii) are immediate from the definition and Theorems 1(iii)-(iv). To verify (iii), observe that if , then it is easily checked that all the -cycles of length 1 are those starting with , whose total number is . Assume now that and that the assertion holds up to . From Theorem 1(vii), the number of -cycles having length is . Thus, to prove the desired result, we only need to verify that Consider Using the induction hypothesis for the second term on the right-hand side of (8) with an extra to take care of the case , we get The result now follows by substituting this into (8).
Implicit in the proof of Theorem 2 is the following nice identity.
Corollary 3. With the same notation as in Theorem 2, one has
2.1.2. The Case
When , more precise information about -cycles is now derived.
Theorem 4. Let . Then (i);(ii) is the largest length among -cycles and any other length divides ;(iii)the number of -cycles having length is .
Proof. Since mod , we have . Assertion (i) thus follows from the observation that . Assertions (ii) and (iii) are direct consequences of Theorems 1(iii)–(vii).
In , Yucas and Mullen introduced the following set: We show next that elements in this set can also be described through order.
Lemma 5. Let , . Then
the multiplicative order of, that is, , and there is a positive integer dividing such that .
Proof. If , then . If , then [2, Proposition 1] tells us that must be an even multiple of . Thus, . Conversely, assume that and is the least positive integer for which . Thus, [2, Proposition 1] tells us that is an even multiple of , and so . If , the divisibility contradicts . Hence, .
The next proposition, which is of independent interest, connects the sum in Theorem 4(iii) with one involving the Möbius function.
Proposition 6. One has
To prove Proposition 6, we need some arithmetical facts about greatest common divisors.
Lemma 7. Let . Then where denotes the highest power of that divides .
Proof. To prove (13), we consider two separate cases.
Case 1 (). Observe that If , then , and we infer that .
If , then , and we infer that .
Case 2 (). Observe that If , then , and from (16), we have On the other hand, Since is odd, we have . This last relation together with (17) and (18) yields , as required.
If , then , and so we infer from (16) that Since (18) holds, we have Since is odd, we also have The relations (20) and (21) show that for some , . Taking (19) into account, we deduce that or . To complete the proof, it suffices to verify that . To this end, suppose that . Thus, (22) yields Since is odd, we see that , where is odd. Combining this last relation with (23), we arrive that , which is untenable.
To prove (14), we first observe that Putting , in (13), the desired result (14) follows at once if we can show that To this end, let , . It is easily checked that because and .
We are now ready to prove Proposition 6.
Case 1 ( is a power of ). Clearly, in this case for . For brevity, let Thus, . If , then using (14), we see that those for which must satisfy , showing that . On the other hand, if , then using (14), we see those for which must satisfy , showing that . Combining the two possibilities, we get
Case 2 ( is not a power of ). Then , where and is odd. From (14), we need to find those for which . These are such that ; that is, . Thus, all such are of the form and correspondingly, , for some , . We proceed now by induction on , noting that the case (and ) is contained in Case 1. We have
3. Order of a Polynomial
Following [3, page 84], for with , the least for which divides is called the order of , denoted by ord. If , then , where and with are uniquely determined, and ord is defined to be ord. The order of a polynomial is also called the period or the exponent of that polynomial. The following facts are well-known.(i)(see [3, Theorem 3.3]) Let be irreducible with , . Then ord is equal to the order of any root of in the multiplicative group .(ii)(see [3, Lemma 3.6]) Let . Then with divides if and only if ord.(iii)(see [3, Theorem 3.8]) Let be irreducible over with and ord, and let with . Let be the smallest integer with , where is the characteristic of . Then ord.(iv)(see [3, Theorem 3.9]) Let be pairwise relatively prime nonzero polynomials over , and let . Then ord.
3.1. Counting Formulae
Working with -cycles, apart from the known result, [3, Theorem 3.5], that the number of monic irreducible , for which ord and is , some new information about the number of monic irreducible polynomials is now derived.
(i) The polynomial constructed via (2) from a -cycle with is a monic irreducible polynomial with , and . Conversely, each irreducible polynomial with , and is an irreducible factor of arisen, through (2), from an -cycle with .
(ii) The set of monic, irreducible , for which is identical with the set of irreducible factors of .
(iii) The number of -cycles, the number of monic irreducible factors of , and the number of monic irreducible with and are all equal to .
Proof. From Theorem 1(v), each -cycle with has length and it thus gives rise, through the -correspondence, to a monic irreducible , , of degree . Each root of is of the form where is a primitive th root of unity, and so its order is . Part (i) is thus a consequence of [3, Theorem 3.3] and the -correspondence.
To prove (ii), note that if the monic irreducible satisfies and ord, then ord for some . Thus, , and so . On the other hand, if is an irreducible factor of , the -correspondence assures us that is constructed from a -cycle and the desired conclusion follows from (i).
Part (iii) follows (ii), the -correspondence, and Theorem 1(viii).
There is another straightforward technique to derive a formula for the number of monic irreducible polynomials of fixed order using -cycles based on the inclusion-exclusion principle. Although the formula so obtained is not easy to use, we give a proof to illustrate this different approach. For , set (keeping the earlier notation in the first section) For a given and distinct primes , define
Proposition 9. Let , the unique prime factorization of . The number of monic, irreducible polynomials , having order is equal to
Proof. Since the number of distinct irreducible factors of in is equal to the number of -cycles [1, Corollary 9.12], from its definition, the -correspondence and [3, Lemma 3.6], is simply the number irreducible factors (in constructed through (2)) of which are not irreducible factors of for any , , which is in turn equivalent to saying that is the number of monic, irreducible polynomials , , having order , and this proves (32).
Again using [1, Corollary 9.12] and [3, Lemma 3.6] to translate the number of -cycles to those of corresponding irreducible polynomials, the inclusion-exclusion formulae on the right-hand sides of (33) and (34) clearly yield the number of irreducible factors of which are not irreducible factors of for all , and this verifies (33) and (34).
Remark 10. Let us remark that there are no irreducible polynomials in of order with . This is seen as follows: for with , since ord is the least positive integer for which divides , monic irreducible polynomials of order are contained in the set of irreducible factors of in . Putting , since , the set of all irreducible factors of is identical with the set of all irreducible factors of , and so each irreducible factor of is of order .
In contrast to the preceding remark, there always exists a (reducible) polynomial in of order with . This is seen as follows: using the terminology of the last remark, since , Theorem 8 assures us that there always exists a monic, irreducible polynomial , , having . Consequently, Theorem 3.8 in  tells us that ord.
Our next task is to compute the number of monic polynomials in (both reducible and irreducible) having order . Let be a factor of the polynomial with ord. Since , the polynomial has no multiple factors. Assume that is decomposed into distinct monic irreducible factors (in ), say, . Since , each ord (the set of all divisors of ). Thus, the number of all such 's is equal to . We have thus proved.
Theorem 11. The total number of monic polynomials (both reducible and irreducible) , , having order is
Having determined the number of polynomials with fixed order, it is natural to find out how many of them have orders dividing a fixed .
Theorem 12. Let with and . Then the product of all monic irreducible polynomials , for which is equal to and their number is equal to .
Proof. Since , the set of all monic irreducible factors of is identical with the set of all monic irreducible factors of . From [3, Theorem 3.6], for , , we know that ord if and only if . Thus, the set of all monic irreducible polynomials , for which ord is identical with the set of all irreducible factors of which is also equal to the set of all irreducible factors of . Since has no multiple root, the product of all monic irreducible polynomials , for which ord is and the first assertion is established. The second assertion follows immediately from the first assertion and Theorem 8(iv).
3.2. Some Explicit Shapes
Having counted the number of monic irreducible polynomials of fixed order, we proceed to determine their explicit shapes. From [1, Corollary 9.12], we know that the number of irreducible factors of in is equal to the number of -cycles formed by the numbers . To determine explicit shapes of all monic, irreducible polynomials , , having order , we consider the polynomial associated with the -cycle, where is a primitive th root of unity. By definition, [3, Lemma ] and [1, Theorem ], among all such polynomials, those which do not divide , , are all the sought after polynomials having order .
Although, the procedure just described is satisfactory in principle, in certain cases, more precise shapes can be given. For a positive integer not divisible by , recall that the th cyclotomic polynomial [3, page 64] is defined as where is a primitive th root of unity and is the Möbius function. It is known that
Proposition 13. Let be the number of divisors of . (i)If the number of -cycles is equal to , then in (38) each is irreducible in and . In particular, is irreducible and of order . The same conclusion holds if is a primitive root mod for all .(ii)If is irreducible in for all , then the number of -cycles is equal to .
Proof. We prove only part (i), for the other part is trivial from (38). If the number of -cycles is equal to , the -correspondence shows that the number of monic irreducible factors of is also . The first assertion that each is irreducible follows at once from (38). This is also the case if is a primitive root mod for all which is a direct consequence of [3, Theorem 2.47(ii)].
From (38), we see that divides . If ord, then divides . Further, Lemma 3.6 of  shows that , and so divides . The first assertion then implies that each cyclotomic factor of and of (as in (38)) is irreducible. Since , the polynomial contains no multiple root and so all its irreducible factors are distinct. But the cyclotomic factorization of (as in (38)) does not contain the irreducible factor (because ), which is a contradiction.
Our next proposition, which shows how the number of -cycles can be used to generate irreducible polynomials, is based on the following known results.(i)(Berlekamp's factorization theorem, [3, Theorem 4.1]). If is monic and is such that , then .(ii)(see [1, Theorem 9.13]) For each -cycle, if , then . Combining these two results, we get the following.
Proposition 14. Let be a -cycle, and . If the number of -cycles is equal to , and if for all , then each polynomial is irreducible over .
4. Degree of a Polynomial
We begin by listing (without proofs) known facts involving degrees of irreducible polynomials, which can be proved using -cycles.(i)(see [3, Theorem 3.20]) The product of all monic irreducible , for which is equal to .(ii)(see [3, Theorem 3.25]) The number of monic irreducible polynomials in of degree is equal to .(iii)(see [3, Corollary 3.4]) The set of monic irreducible , for which is identical with the set of monic irreducible , for which ord. Moreover, for , , we have if and only if ord. Using -cycles, that is, Theorem 1(viii) and the assertion [3, Theorem 3.20] the following new result is immediate.
Theorem 15. The number of monic irreducible polynomials , for which is equal to , where the definition of can be found in the heading “Notation and Terminology”.
Corollary 16. One has .
5. Coefficients and -Cycle
Since a given -cycle gives rise, through the -correspondence, to a unique polynomial, we illustrate now how to uniquely determine all coefficients of its corresponding polynomial (2) and conversely.
Theorem 17. Let be a primitive th root of unity, let be the order of , and let be a generator of .(i)If the -cycle is given, then the coefficients of its corresponding polynomial in (2) are given by (ii)If the coefficients of are given, then the corresponding -cycle is uniquely determined through its leading element which is found by solving for , , and , respectively, from the system
Proof. (i) The relations in (39) are simply the symmetric polynomial relations between the roots and coefficients of a polynomial.
(ii) It is trivial that each -cycle is uniquely determined from its leading element. Next, note that (40) is simply a representation of element in . Through the -correspondence, the -cycle associated with must satisfy The conclusion follows from the facts that and that solving (41) and (42) yields uniquely mod .
6. Primitive and Self-Reciprocal Polynomials
We now apply preceding results to two special cases, corresponding to and , to deduce a number of results about primitive and self-reciprocal polynomials.
6.1. Primitive Polynomials
Recall from [3, Definition 3.15] that a primitive polynomial of degree over is the minimal polynomial over of a primitive element of . We mention, without proofs, some known results provable using -cycles.(1)(see [3, Theorem 3.16]) An irreducible polynomial of degree is primitive if and only if is monic, and ord. In other words, each primitive polynomial in of degree can be constructed from a -cycle with through (2). Conversely, for each -cycle with , the polynomial (2) is a primitive polynomial in of degree .(2)(see [1, Theorem 7.7]) The number of primitive polynomials in of degree is , which equals to the number of -cycle with .
6.2. Self-Reciprocal Polynomials
Let with . The reciprocal polynomial of is defined by . A nonzero polynomial is called self-reciprocal if . We confine ourselves here to the study of self-reciprocal irreducible monic (srim) polynomials in . Since there is only one first degree srim-polynomial, namely, , throughout the rest of this section we treat only srim-polynomials of degree ≥2. We next mention a known characterization and results about srim-polynomials provable via -cycles.(i)(see [2, page 275]) Let be irreducible and monic of degree ≥2. Then is self-reciprocal if and only if its set of roots (each of which is evidently non-zero) is closed under inversion (and so its degree must be even).(ii)(see [2, Proposition 3]) If is a srim-polynomial with , then ord but ord for all .(iii)(see [4, Lemma 2.3], [5, Theorem 1(i)], [2, Corollary 5]) For with , each -cycle, which is of length , gives rise through (2) to a srim-polynomial in of degree and order dividing . Conversely, each srim-polynomial of degree arises via (2) from a -cycle with length and ord, where .(iv)(see [5, Theorem 1(ii)]) Each irreducible factor of with is a srim-polynomial with , where and is odd. We next prove two new results using -cycles.
(i) For each divisor of , the number of all srim-polynomial's for which and is .
(ii) The number of srim-polynomials of degree in is equal to .
Combining [2, Proposition 3], [5, Theorem 1(i)-(ii)], and Theorem 18 with results in Section 2, we obtain the following known results about the set as defined in (11).(i)(see [2, Proposition 4]) Assume that is a srim-polynomial over of degree and let be a root of . Then is a primitive th root of unity for some .(ii)(see [3, Exercise 3.15, page 141]) If is a srim-polynomial in of degree >1 and order , then every monic irreducible polynomial in of degree >1 whose order divides is self-reciprocal.(iii)(see [2, Corollary 5], [6, Theorem 2]) If is a srim-polynomial of degree over , then ord.(iv)(see [2, Proposition 7]) Let and let be a primitive th root of unity. If , then the polynomial is an srim-polynomial of degree and order .(v)(see [2, Theorem 8]) For a monic irreducible polynomial of degree over , the following statements are equivalent.(1) is self-reciprocal.(2)ord.(3) for some primitive th root of unity , with .(vi)(see [2, Theorem 9])(1)There are srim-polynomials in of degree and order for each .(2)The number of srim-polynomials in of degree is